ABSTRACT
Mathematical study of human pulse wave was studied with the view to gaining an insight into
physiological situations. Fluid –Structure interaction (FSI) in blood flow is associated with
pressure pulse wave arising from ventricular ejection. Solution of the coupled system of nonlinear
PDEs that arose from the FSI was sought in order to determine pressure. Further study on
pressure pulse waves showed that the Korteweg-de Vries (KdV) equations hold well for the
propagation of nonlinear arterial pulse wave. Solutions of the KdV equation by means of the
hyperbolic tangent (tanh) method and the bilinear method each yielded solitons. The solitons
describe the peaking and steepening characteristics of solitary wave phenomena.
The morphologies of the waves were studied in relation to the length occupied by the waves
(which corresponds to length of arterial segment and stature) and the left ventricular ejection
time (LVET). The study showed that both stature and LVET are independent descriptors of
cardio-vascular state.
TABLE OF CONTENTS
Title page
Certificate of Approval i
Acknowledgement ii
Dedication v
Abstract vi
Chapter One
Overview of Fluid-Structure Interaction Problem 1
1.0 Introduction 1
1.1 Body vessels 1
1.2 Arteries and their structure 4
1.3 Blood 6
1.4 Blood Pressure (BP) 7
1.5Arterial Pulse 8
1.6 Pulse Pressure 9
1.7 Fluid-Structure Interaction 10
1.8 Pulse Wave 11
1.9 Resonance 12
1.10 Aim and objectives of the study 12
1.11 Scope and limitations of the study 13
1.12 Methodology 13
1.13 Significance of the study 14
Chapter Two
Literature Review 15
ix
Chapter Three
Fluid-Wall interaction and non-linear pulse wave models in blood flow 21
3.1.0 Generalized equation of motion of viscous fluid 21
3.1.1 Action of fluid on the wall 26
3.1.2 Fluid –Structure interaction model: Problem presentation 29
3.1.3 Fluid –Structure coupling 36
3.2.0 Model of non-linear arterial pulse: Problem presentation 39
3.2.1 Linear superposition of forward and backward ABP waves 41
Chapter Four
Solutions to model problems 44
4.1.0 Solution of fluid-wall interaction problem 44
4.1.1 Weak Formulation and Variational Form 45
4.1.2 Rescaled Problem and asymptotic expansion 50
4.1.3 Weak Formulation 51
4.1.4 Energy estimates after rescaling 52
4.1.5 Asymptotic Expansions 53
4.1.6 Justification for asymptotic expansions 54
4.1.7 Reduced problem using Expansion I 55
4.1.8. Reduced problem using Expansion II 58
4.2.0. Nonlinear arterial pulse model 62
4.2.1 Methods of Solution of non-Linear Wave model problem 67
4.2.2 The Tanh (hyperbolic tangent) Method of Solution 68
4.3.0 Bilinear Method 73
4.3.1 Solitons by bilinear method 75
4.4 Systolic and Diastolic PW Representation 80
Chapter Five
Results and discussions 83
5.1.0 Features of: 84
x
5.1.1 Tanh method 84
5.1.2 Bilinear Method 84
5.2.0 Physiological Analysis using solitary waveform 85
5.2.1Distance effect 85
5.2.2 Short and tall statures 86
5.2.3 Time effects 91
5.2.4 Harmonic Components of Arterial Pulse Waves 94
5.2.5 Heart-Organ Resonance 95
5.2.6 Hypertension and vaso-active Agents 96
5.2.6 Dying Process 98
5.3.0 Summary and Conclusion 99
5.4.0 Recommendation(s) for further studies 101
References 102
Appendices 112
1
CHAPTER ONE
OVERVIEW OF FLUID-STRUCTURE INTERACTION
1.0 Introduction:
In this work we analyzed hemodynamic pulse waves (PW) in human fluid-structure
interaction problems. The work engaged mathematical models to show, among other things,
that arterial pressure which has systolic and diastolic components generates PW which are
enough to determine the physiological state of each of the internal organs, especially of the
heart. The understanding of some of the terms used in this work may be necessary. In
subsection 1.1 below some of such terms are explained.
1.1 Body vessels
In anatomy, a vessel is a tubular structure that conducts body fluid: a duct that carries fluid,
especially blood or lymph to parts of the body. Thus, blood vessels are blood-carrying ducts.
Blood vessels are in three varieties: arteries, veins and capillaries.
Arteries
The main arteries are:
Pulmonary arteries: Carry deoxygenated blood from the body to the lungs where it is
oxygenated and freed of carbon dioxide.
Systemic arteries: They deliver blood to the arterioles, then to the capillaries where gases
and nutrients are exchanged.
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Aorta: This artery is supplied with blood from the left ventricle of the heart
via the aortic valve. It is the root systemic artery, and it branches to daughte arteries. It carries
blood away from the heart.
Arterioles: These are the smallest of the true arteries. They regulate blood pressure and
deliver blood to the capillaries.
Carotid, subclavian, mesenteric, renal, iliac arteries and the celiac trunk are branches of
the aorta
Venules are the small blood vessels that transfer blood from the capillaries to the veins.
Veins
– They are large collecting vessels, such as the subclavian, jugular, renal, and
iliac veins. They carry blood at low pressures.
– Venae cavae are the largest veins, which they carry blood into the heart.
Capillaries
These are the smallest blood vessels (about 5-10μm in diameter).They form part of
microcirculation. Arteries divide into arterioles and continue to narrow, and as they reach the
muscles they become capillaries. Capillaries do not transport blood. They are specially
designed for the passage of substances, mainly oxygen and carbondioxide. They are thinwalled
and are composed only of endotheliac cells, which allow easy passage of substances.
A notable feature of capillary beds is their control of blood flow through auto regulation. This
helps an organ to maintain constant flow despite changes in central blood pressure. New
capillaries can be formed by pre-existing capillaries in a process called angiogenesis. Fig.1.1
shows the location of various arteries in the body.
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Fig1.1 Human arterial system. (Available at www.chakras.org.uk/chakra_yoga_health_holistic_arterial.gif)
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# Artery Name Function
0 Arterial system Canals that carry blood from the heart to the organs
1 Posterior Vessel that carries blood to the head.
3 External carotid Neck vessel that carries blood to the face
4 Internal carotid Neck vessel that carries blood to the brain
5 Common carotid (left) Carries blood to the left side of the neck
6 Brachio- cephalic Main vessel of the arm
7 Left subclavian Carries blood beneath the left clavicle
8 Right coronary artery Feeds the tissues of the right side of the heart with blood
9 Thoracic aorta Main artery of the thorax
10 Celiac trunk Carries blood to the thoracic cavity
11 Renal Carries blood to the kidneys
12 Superior mesenteric Carries blood to the upper part of the abdomen
13 Abdominal aorta Main artery in the abdominal area
14 Inferior mesenteric Carries blood to the lower part of the abdomen.
15 Common illiac Principal artery of the human lower limb
16 Internal iliac Internal branch of the iliac artery
17 External iliac External branch of the iliac artery
18 Profunda femoris Carries blood towards the inside of the thigh
19 Peroneal Carries blood to the lower leg
20 Lateral planter Carries blood to the side of the sole of the foot
21 Dorsalis pedis Carries blood to the dorsal part of the foot
22 Plantar arc Carries blood to the instep area
23 Medial plantar Carries blood to the median of the sole of the foot
24 Anterior tibial Carries blood to the front part of the lower leg
25 Posterior tibial Carries blood to the back part of the lower leg
26 Popliliteal Carries blood to the back of the foot
27 Femoral Carries blood to the thigh
28 Superficial palmar arch Situated beneath the skin of the palmar arc of the hand
29 Ulnar Situated in the area of the ulnar
30 Common interosseous Situated between the two bones of the forearm
31 Gonadal (Genital) Carries blood to the genital organs
32 Radial Situated in the area of the radius
33 Brachial Carries blood to the arm
34 Profunda brachial Carries blood towards the interior of the arm
35 Axillary Carries blood to the armpit
36 Right subclavian Carries blood beneath the right clavicle
37 Right vertebral Situated on the right, carries blood to the vertebrae
38 Common carotid (right) Carries blood to the right of the neck
39 Superior thyroid Carries blood to the thyroid
40 Lingual Carries blood to the tongue
41 Facial Carries blood to the face
42 Maxillary Carries blood to the maxillae
43 Superficial temporal Carries blood to the surface of the skin, in the area of the temples
Table 1.1 The 43 human main arteries and related function (Almanasreh (2007))
1.2 Arteries and their structure
All relatively large arteries have similar basic structure. The artery consists of the outermost
layer known as the tunica adventitia. This layer is composed of connective tissue. The inner
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layer is known as the tunica media, and is made of smooth muscle cells and elastic tissues.
The innermost layer is known as the tunica intima. This layer is in direct contact with the
flowing blood. The lumen is the hollow internal cavity in which the blood flows (as seen in
Fig. 1.2).
Fig 1.2 Photomicrograph of the cross section of an artery showing the tunica intima, tunica media, and tunica externa.
Available at http://www.mhprofessional.com/product.php?isbn=0071472177
The endothelium of the intima is surrounded by sub-endothelial connected tissue. Around this
there exists a layer of vascular smooth muscle, which is developed in arteries. There is a
further layer of connective tissue known as the adventitia, which contains nerves that supply
blood to the muscular layer as well as nutrient to the capillaries in the larger blood vessels.
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Blood vessels do connect and form anastamosis (a region of diffuse vascular supply). In
event of blockages anastamoses provide critical alternative route for the flow of blood.
In course of blood circulation arteries mainly carry blood away from the heart. The
capillaries link the arteries to the veins, and the veins carry the blood back to the heart.
Besides blood circulation, arteries (and blood vessels as a whole) help to measure vital health
statistics such as pulse and blood pressure. We can measure heart rate, or pulse, by touching
an artery. The rhythmic contraction of the artery as the heart beats keeps pace with the pulse.
The proximity of the artery to the surface of the skin enhances the accurate measurement of
the heart’s pulse by touching the artery. The heart itself is deeply protected.
1.3 Blood
Talking about whole blood, we think of the formed elements that are suspended in plasma.
The red blood cells (RBCs) constitute major part of the formed elements. The ratio of this
part to the other constituents of whole blood is known as hermatocrit.
Fig1.3 Formed elements of blood, Dugdale (2010)
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It is the preponderance of the RBCs in the whole blood composition that make them very
important in determining the flow characteristics of blood. RBCs aggregate at low shear rates
(values < 100dyn s-1) and form rouleaux. This has the effect of increasing the viscosity of
blood. Rouleaux disaggregation occurs as shear rate increases, resulting in the shear-thinning
characteristics that cause the non-Newtonian behavior of blood. In effect, blood viscosity
decreases. We consider blood as Newtonian when its shear-thinning characteristic disappears
as a result of increase in shear rate beyond the low shear rate region (Ku (1999), Kang
(2002)).
1.4 Blood Pressure (BP)
Blood pressure is the force being exerted on the walls of the arteries in the event of blood
being transported to parts of the body. It is customary to use the blood flowing from the
arteries to measure blood pressure because it is transported at a higher pressure than the blood
in the veins. BP is measured using two numbers (see Blood pressure chart in Appendix A).
The first number, which is usually higher, is taken when the heart beats during systole (the
contraction of the heart during which blood is pumped into the arteries), as the heart rests
between cycles.The systolic pressure is the peak pressure during heart contraction while the
second number is taken when the heart relaxes during diastole (rhythmic expansion of the
heart’s chambers at each heartbeat during which they fill with blood). The diastolic pressure
is the minimum pressure between contractions. Each of the numbers is recorded in
millimeters column of mercury (mmHg). It is normal for BP to increase in course of exercise
and to decrease when asleep. If BP stays too high or too low, there may be the risk of heart
disease.
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The heart pumps blood out through a main artery known as the dorsal aorta. This main aorta
divides and branches out into several smaller arteries so that each region of the body has a
system of arteries that supply it with fresh oxygenated blood.
When the heart beats (during systole) the artery is filled with blood and it expands. When the
heart relaxes (during diastole) the artery contracts and exerts force that would push the blood
along. The integrity of blood flow and efficient circulation is the synergy between the heart
and the artery.
1.5 Arterial pulse
Pulse may be explained in terms of regular beat of blood flow as the regular expansion of an
artery, caused by the heart pumping blood through the body, or in terms of single beat of
blood flow as a single expansion and contraction of an artery, caused by a beat of the heart.
Usually, in medicine, pulse is the tactile arterial palpation by trained fingertips. Such
palpation may be in any place where an artery can be compressed against a bone. Such places
include neck (carotid artery), the wrist (radial artery), behind the knee (popliteal artery), on
the inside of the elbow (brachial artery), and near the ankle joint (posterior tibial artery), as
shown in Fig.1.4.
Pulse may be used in expediency as a tactile method of determination of systolic blood
pressure to a trained observer, but this cannot be said about diastolic blood pressure. Below
are the physiological pulse rates at rest (the resting heart rate (HRrest) is a person’s heart rate
when they are at rest, that is lying down but awake, and not having recently exerted
themselves http://en.wikipedia.org/wiki/Heart_rate#At_rest).
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newborn
(0-3 months
old)
infants
(3 — 6
months)
infants
(6 — 12
months)
children
(1 — 10
years)
children over 10
years
& adults, including
seniors
welltrained
adult
athletes
100-150 90–120 80-120 70–130 60–100 40–60
Table 1.2 Normal pulse rates at rest, in beats per minute (BPM):US(20
Fig1.4 Arterial pulse points (at: http://www.nakedscience.org/mrg/AnatomyLectureNotesUnit7CirculatorySystem-
TheBloodVessels_files/image006.gif)
1.6 Pulse Pressure (PP)
This is the amount of pressure that is required to create the feeling of a pulse.The variation of
pressure within the artery produces pulse which is transmitted through the artery; hence the
name arterial pressure pulse .When the heart’s left ventricle contracts, systemic arterial
pressures are generated. PP is most easily defined as being the amount of pressure required to
create the feeling of a pulse. The amount of pulse is created is measured (in mmHg) by the
Systolic versus Diastolic difference of blood pressures. It is mainly related to the amount of
blood ejected by each heart beat, stroke volume and the elasticity of the major arteries. If you
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have a resting blood pressure is (systolic/diastolic) 120/80 mmHg, then your PP is 40, which
is a healthy PP.A consistent high resting PP is very harmful, and is most likely to quicken the
normal ageing of the heart, brain and kidney. In recent times researches have been carried out
to determine the relationship between pulse pressure and hypertension (Martin et al (1995)).
There is the possibility that resistance vessel structural adaptation in hypertension may be
closely related to pulse pressure than to other blood pressure parameters. Studies show that
the widening of PP is seen as a consequence of a loss of compliance in the large conduit
arteries and increased wave reflections from the periphery, and less to increased resistance in
peripheral arteries (Safar (1993)).It is most likely to be evident in, and prognostic of, cardiovascular
abnormally.
1.7Fluid-Structure Interaction (FSI)
The notion of fluids-structure interaction with regard to human dynamical structure is the
interplay of blood flow and the arterial conduit. The flow of blood via the arterial conduit
induces concomitant motion of the arterial wall, and thus there exists some interfacial region
within which fluid-structure interaction (FSI) is felt. Flow usually occurs in radial, angular
and axial directions. The arterial wall is expected to be compliant in order to sustain the
integrity of the flow. In compliant arteries flow in radial direction induces radial dilatation,
whilst axial flow induces longitudinal (axial) stretch.
In the main, the circulatory system of animated life owes its integrity to FSI. Within
reasonable expectation this interaction would present some mixed blessing to human
physiological state. Within physiological range of flow and pressure, the body would be held
at normal cardio-vascular state. On the unpleasant version, there could be incipient pathology
in the physiological state when the interaction fails to yield the desired goal. Beside various
hemodynamic para- meters we take exceptions at arterial pressure pulse wave. We shall give
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attention to details on the nonlinear waves generated by the pulse. The injection of blood
from the left ventricle into the aorta generates far more than traveling waves in the arterial
system, otherwise forced stationary oscillations of the entire arterial system would be
untenable; thus pressure pulse would not likely reveal details about the resonance conditions
of the whole body in a defined manner. It is against this background that we are attracted to
the mathematical analysis of pulse waves and the resonance conditions there from, in a bid to
furnish medics with some contribution to wave-related issues in physiology. As we attest to
the solitary nature of the pulse waves, Wang et al (2010) reminds us in more physiological
terms that the resonance conditions of the whole body, sequel to the wave train must be a
benchmark for normal conditions of organs. This reminder is worthwhile in treating the
waveforms alongside their harmonic components. With this treatment, salient information
could be supplied regarding organ patho-physiology and hypertension. In this regard medics
could be availed of the benefits derivable from this work.
1.8 Pulse Wave (PW)
Pulse wave is a very complex physiological phenomenon observable, and detectable in blood
circulation. Any segment of blood ejected and transported through the artery in event of heart
systole is transformed between kinetic and potential energy. There are three observable
coherent phenomena on each artery or venous segment affected by pulse wave: blood flow
(flow pulse), the change in BP (pressure pulse) and the extension in transverse profile
(volume pulse). PPW is generated from the combination of the incident wave (i.e. the
pressure wave generated by the left ventricle in systole) and waves reflected back from the
periphery.
The shape of PW changes as moves to the distal arteries. This change could be largely due to
the pulse wave reflection and arterial tapering.
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Pulse waveform varies in different vessels in the same individual. It is depends on the (viscoelastic)
properties of the artery (inducing wave amplification as it travels from more elastic
central to stiffer peripheral arteries), the viscosity of the blood, wave reflection and wave
dispersion.
1.9 Resonance
The term resonance as used here is in association with mechanical systems. In this regard it is
used to describe large oscillation at natural frequency. In the event of physiological arterial
flow, there is increased amplitude of oscillation of the organs (the mechanical system), when
they are subjected to vibration arising from the PW.The wave transmits through the artery
(source) proximal to the organ in question. Each of the organs (specifically the heart, liver,
lungs, kidney and spleen) has its own natural frequency. Each of the organs is influenced by
both the input pulsatile blood and by the harmonic driving force, both of which are
proportional to the PP and the connecting site of the local main artery. PP is good enough to
reveal information about the resonance conditions of the whole body. Wang et al (1990),
therefore was of the view that periodic injection of blood from the left ventricle would not
generate only traveling waves, but would also induce forced stationary oscillation of the
arterial system. This work buys this view, and would use the organ specific harmonics to
analyze the patho-physiological state of the body.
1.10 Aim and objectives of the study
The aim of this work is to determine whether stature (height) and LVET each can be
implicated in cardiac-vascular events.
The objectives of this work are:
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To carry out a study of hemodynamic pressure pulse wave due to left
ventricular ejection.
To use pulse waveforms to analyze the human physiological state.
To extend the frontiers of existing knowledge of the subject under consideration.
1.11 Scope and limitations of the study
This work studied hemodynamic pulse waves arising from the heart’s left ventricular ejection
and the FSI. It studied the effect of stature on pulse waveforms and analyzed the
physiological implications of the waveforms.
The study was non-invasive (as applicable in most medical practice). However, the work
relied on some experimental findings where necessary. We must agree, without prejudice,
that it takes maturity to know that models should be used, not believed (Henry (2012)).
1.12 Methodology
A set of nonlinear partial differential equations was used to describe the pressure-induced
viscous flow of the fluid. Another set of nonlinear partial differential equations was used to
describe the motion of the arterial wall. We expressed the FSI as a coupled system and the
equation of the radial contact force was obtained. The model of arterial pulse was built on the
system of equations governing the FSI. Analytic method was used to get the solutions of the FSI
problem which yielded the desired equation of arterial pressure. In a similar way, the solution of
the arterial pulse was sought. We used a combination of the hyperbolic tangent (tanh) and the
bilinear methods of solution to get the solutions of the arterial pulse problem. We obtained
solitary wave (soliton-) solutions. We therefore described pulse waves as solitons. Matlab
Software was used for the processing and plotting of graphs of our solutions and our analysis.
We went further to analyze the physiology underlying the pulse waves. Stature and the LV
14
ejection time were the parameters used for clinical purposes. In each of the cases the values of
the parameters were varied in order to study the (patho-) physiology underlying arterial pulse
waveforms.
1.13 Significance of the study
This work is important in many respects:
(i) It would underscore the importance of arterial pulse waveforms in determining pathophysiological
state of humans.
(ii) It would help medics to decide the course of their treatment to related pathologies.
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